Discrete Mathematics

Fibonacci Numbers

Challenge Quizzes

Fibonacci Numbers: Level 3 Challenges

         

Compute 12+14+28+316+532++Fk2k+ \frac{1}{2} + \frac{1}{4} + \frac{2}{8} + \frac{3}{16} + \frac{5}{32} + \dots + \frac{F_k}{2^k} + \dots where Fk F_k represents the the kth k^\text{th} term of the Fibonacci sequence: 1,1,2,3,5,8,13,...1, 1, 2, 3, 5, 8, 13, ...

Find the last digit of the 123456789-th Fibonacci number.

In the Fibonacci sequence, F0=1F_{0}=1, F1=1{F_1}=1, and for all N>1N>1, FN=FN1+FN2F_N=F_{N-1}+F_{N-2}.

How many of the first 2014 Fibonacci terms end in 0?

The Fibonacci sequence is defined by F1=1,F2=1F_1 = 1, F_2 = 1 and Fn+2=Fn+1+Fn F_{n+2} = F_{n+1} + F_{n} for n1 n \geq 1 .

Find the greatest common divisor of F484F_{484} and F2013F_{2013}.

n=0Fn3n= ?\sum_{n=0}^{\infty} \frac{F_{n}}{3^{n}}= \ ?

Details and Assumptions

FnF_{n} is the nthn^\text{th} Fibonacci number, with F1=F2=1F_{1} = F_{2} = 1.

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